The Generalized Tonnetz Dmitri Tymoczko Abstract This article relates two categories of music-theoretical graphs, in which points represent notes and chords, respectively. It unifies previous work by Brower, Callender, Cohn, Douthett, Gollin, O’Connell, Quinn, Steinbach, and myself, while also introducing new models of voice-leading structure—including a three-note octahedral Tonnetz and tetrahedral models of four-note diatonic and chromatic chords. !"#$% &'()*$#&# &+,$%-..+ *(,*(#(/& voice leading using two different kinds of diagram. In note-based graphs, points represent notes, and chords cor- respond to extended shapes of some kind; the prototypical example is the Tonnetz, where major and minor triads are triangles, and where parsimoni- ous voice leadings are reflections (“flips”) preserving a triangle’s edge. In chord-based graphs, by contrast, each point represents an entire sonority, and efficient voice leading corresponds to short-distance motion in the space, typically along an edge of a lattice. This difference is illustrated in Figure 1, which offers two perspectives on the same set of musical possibilities: on the top, we have the traditional note-based Tonnetz, while on the bottom we have Jack Douthett and Peter Steinbach’s (1998) chord-based “chicken-wire torus.”1 These figures both represent single-step (or “parsimonious”) voice leading among major and minor triads and are “dual” to each other in a sense that will be discussed shortly. In A Geometry of Music (Tymoczko 2011), I provide a general recipe for constructing chord-based graphs, beginning with the continuous geometri- cal spaces representing all n-note chords and showing how different scales determine different kinds of cubic lattices within them. I also showed that nearly even chords (such as those prevalent in Western tonal music) are rep- resented by three main families of lattices. Two of these are particularly use- ful in analysis: the first consists of a circle of n-dimensional cubes linked by Thanks to Richard Cohn and Gilles Baroin for helpful comments. 1 The chicken-wire torus was introduced in Douthett and Steinbach 1998. There are many different orthographical variants of the traditional Tonnetz, depending on how one orients the axes; for a survey, see Cohn 2011a. For present purposes, these are all equivalent. Journal of Music Theory 56:1, Spring 2012 DOI 10.1215/00222909-1546958 © 2012 by Yale University 1 2 JOURNAL of MUSIC THEORY (a) (b) Figure 1. Two versions of the Tonnetz. (a) The note-based version, in which points represent notes and triangles represent chords. (b) Its geometrical dual, called the “chicken-wire torus” by Douthett and Steinbach (1998). Here, points represent chords and edges represent single-step voice leading. Dmitri Tymoczko The Generalized Tonnetz 3 shared vertices; the second consists of a circle of n-dimensional cubes linked by shared facets (the third does not often appear in practical contexts, and can be safely ignored).2 What results is a systematic perspective on the full family of chord-based graphs. The question immediately arises whether we can provide a similarly systematic description of the note-based graphs. What note-based construc- tion represents efficient voice leading among nearly even four-note chords in the chromatic scale? What about nearly even four-note chords in the diatonic scale? How can we generalize the familiar Tonnetz to arbitrary chords within arbitrary scales? Is there a note-based graph for every chord-based graph? Is one or the other type of graph more useful for particular applications? The purpose of this article is to answer these questions by providing a recipe for constructing generalized note-based graphs, or Tonnetze. Along the way we will encounter some surprising facts: • The Tonnetz, while apparently a two-dimensional structure, can also be understood as a three-dimensional circle of octahedra linked by shared faces. The shared faces represent augmented triads, which do not appear on the traditional Tonnetz. The two versions of the Ton- netz are graph-theoretically identical but geometrically (and topo- logically) distinct. • The seventh-chord analogue to the traditional Tonnetz can be depicted as a series of nested tetrahedra, each containing the notes of a diminished-seventh chord. This figure represents efficient voice leading among diminished, half-diminished, dominant seventh, minor-seventh, and French sixth chords. • The traditional Tonnetz is often described as a torus, or a “circle times a circle.” However, the more general description is the “twisted product of an (n – 2)-dimensional sphere with a circle.” It just so hap- pens that in the three-note case, the one-dimensional sphere is itself a circle, potentially misleading theorists into thinking that higher- dimensional Tonnetze are also toroidal. • Any sufficiently large note-based graph will inevitably contain either “flip restrictions” or “redundancies”—that is, the graph will either contain “flips” that represent nonstepwise voice leadings or multiple representations of the same chord. The traditional Tonnetz is unusual in that it lacks both features. • Chord-based voice-leading graphs are associated with note-based Tonnetze by the geometrical property of duality. However, the duality 2 This type of graph occurs only when the number of ter these graphs, for scales smaller than fourteen notes, is notes in the chord is less than half the size of the scale, and when exploring four-note chords in a ten-note scale, hardly shares a common factor with it, but does not divide the an everyday occurrence. size of the scale exactly. The only time we would encoun- 4 JOURNAL of MUSIC THEORY relation obtains not between graphs considered as unified wholes, but rather between their cubic and octahedral components. From a theoretical point of view, the last point is the crucial one. The most natural route to the generalized Tonnetz begins with the chord-based lat- tices described in A Geometry of Music. These are typically arrangements of n-dimensional cubes. We can replace each n-cube with its geometrical dual, producing a collection of “generalized octahedra.” These generalized octa- hedra then need to be rotated or reflected before they can be glued together to form the note-based analogue to the original chord-based graph. Geometrical investigations of chordal voice leading began with the note-based Tonnetz, a structure that was originally devised to represent purely acoustical relationships among notes.3 But as the geometrical approach matured, it gradually moved toward chord-based graphs, which are more eas- ily generalized to a broader range of musical circumstances. Having under- stood these chord-based structures, we can now complete the circle, return- ing to the note-based graphs that started the investigation. Thus, more than two decades after the beginnings of neo-Riemannian theory, we are poised to understand the Tonnetz in a deeper and more principled way. 1. Mathematical Background This section reviews some basic mathematical material, beginning with ele- mentary geometrical terminology and proceeding to describe the duality of the hypercube and the cross-polytope. I will try to be informal and intuitive, in keeping with my goal of remaining comprehensible to readers who are musicians first and foremost. This is consistent with my philosophy that music theory is an applied discipline in which mathematics is a tool rather than an end in itself.4 One word of warning: when doing higher-dimensional geometry, it is often necessary to prioritize algebra over direct visualization. In large part, geometry is a matter of grasping patterns that repeat themselves in increas- ing dimensions, with algebraic relations providing our best guide as to which properties do, in fact, generalize. Thus, rather than struggling to construct a vivid picture of the seven-dimensional cross-polytope, one should instead concentrate on the generic properties shared by all cross-polytopes, content- ing oneself with visualizing only the lower-dimensional examples.5 That said, music can sometimes be a useful tool for picturing higher-dimensional rela- 3 It was Richard Cohn (1997) who pioneered the use of 4 Readers who pine for mathematical rigor will likely be able geometrical graphs, and in particular the note-based Ton- to generate proofs from the following informal exposition. netz, to represent chordal voice leading. Some earlier work, 5 A substantial number of blind mathematicians are geom- such as Roeder 1984 and 1987, used geometry to repre- eters (Jackson 2002). One hypothesis is that blindness can sent voice leading among set-classes. For a brief history of be helpful, insofar as it reduces the reliance on quasi-visual the development of geometrical models of voice leading, pictures. see Tymoczko forthcoming. Dmitri Tymoczko The Generalized Tonnetz 5 tionships; for example, readers who have finished this article will have no trouble imagining the seven-dimensional cross-polytope as a certain collec- tion of relations between two completely even seven-note scales. Basic terminology In plane Euclidean geometry, a polygon is a two-dimensional plane figure bounded by a closed sequence of line segments. A vertex is a point belonging to two adjacent line segments. A polygon is said to be convex if its interior contains every line segment between any two points of the polygon. (Convex polygons have internal angles less than or equal to 180°.) These definitions can be generalized to higher dimensions: the three-dimensional analogue of a polygon is a polyhedron, while the n-dimensional analogue is a polytope. A polyhedron is bounded by polygonal faces (dimension 2) that intersect at lin- ear edges (dimension 1), which in turn intersect at points called vertices (dimen- sion 0). An n-dimensional polytope is bounded by facets that are all (n – 1)- dimensional polytopes, themselves intersecting to form (n – 2)-dimensional polytopes (ridges) that intersect to form (n – 3)- dimensional polytopes (peaks) . all the way down to two-dimensional faces, one-dimensional edges, and zero- dimensional vertices.